Description
In this research, the use of Bi-Level Forward Difference (BLFD) for cubic spline interpolation is proposed and the bounds of the interpolant and spline coefficients are determined. As a part of the interpolation process, cubic polynomials must be evaluated at a large number of points on a uniform grid to generate the interpolated values. Using Forward Differences, the work required for function evaluation is greatly reduced, but at the cost of significant accumulation of error. This accumulation of error makes Forward Difference unappealing for all but the shortest splines. BLFD is a novel approach which uses double Forward Differences to reduce the accumulated error and greatly improve the maximum spline length. BLFD reduces the growth of the bound of the error due to accumulation below the growth of the bound of the error due to finite-length calculation of the polynomial coefficients. BLFD is found to be suitable for an efficient, fixed-point hardware implementation of cubic spline interpolation. Additionally, the worst-case bounds of the cubic spline coefficients and the interpolated function are found for the general case.
